Author | Knorr, Wilbur Richard. author |
---|---|
Title | The Evolution of the Euclidean Elements [electronic resource] : A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry / by Wilbur Richard Knorr |
Imprint | Dordrecht : Springer Netherlands, 1975 |
Connect to | http://dx.doi.org/10.1007/978-94-010-1754-1 |
Descript | XI, 379 p. online resource |
I / Introduction -- I. The Pre-Euclidean Theory of Incommensurable Magnitudes -- II. General Methodological Observations -- III. Indispensable Definitions -- II / The Side and the Diameter of the Square -- I. The Received Proof of the Incommensurability of the Side and Diameter of the Square -- II. Anthyphairesis and the Side and Diameter -- III. Impact of the Discovery of Incommensurability -- IV. Summary of the Early Studies -- III / Platoโs Account of the Work Of Theodorus -- I. Formulation of the Problem: ????ยต??? -- II. The Role of Diagrams: ??????? -- III. The Ideal of Demonstration: ?????????? -- IV. Why Separate Cases? -- V. Why Stop at Seventeen? -- VI. The Theorems of Theaetetus -- VII. Theodorasโ Style of Geometry -- VIII. Summary of Interpretive Criteria -- IV / A Critical Review of Reconstructions of Theodorusโ Proofs -- I. Reconstruction via Approximation Techniques -- II. Algebraic Reconstruction -- III. Anthyphairetic Reconstruction -- V / The Pythagorean Arithmetic of the Fifth Century -- I. Pythagorean Studies of the Odd and the Even -- II. The Pebble-Representation of Numbers -- III. The Pebble-Methods Applied to the Study of the Odd and the Even -- IV. The Theory of Figured Numbers -- V. Properties of Pythagorean Number Triples -- VI / The Early Study of Incommensurable Magnitudes: Theodorus -- I. Numbers Represented as Magnitudes -- II. Right Triangles and the Discovery of Incommensurability -- III. The Lesson of Theodorus -- IV. Theodorus and Elements II -- VII / The Arithmetic of Incommensurability: Theaetetus and Archytas -- I. The Theorem of Archytas on Epimoric Ratios -- II. The Theorems of Theaetetus -- III. The Arithmetic Proofs of the Theorems of Theaetetus -- IV. The Arithmetic Basis of Theaetetusโ Theory -- V. Observations on Pre-Euclidean Arithmetic -- VIII / The geometry of incommensurability: Theaetetus and Eudoxus -- I. The Theorems of Theaetetus: Proofs of the Geometric Part -- II. Anthyphairesis and the Theory of Proportions -- III. The Theory of Proportions in Elements X -- IV. Theaetetus and Eudoxus -- V. Summary of the Development of the Theory of Irrationals -- IX / Conclusions and Syntheses -- I. The Pre-Euclidean Theory of Incommensurable Magnitudes -- II. The Editing of the Elements -- III. The Pre-Euclidean Foundations-Crises -- Appendices -- A. On the Extension of Theodorasโ Method -- B. On the Anthyphairetic Proportion Theory -- A List of the Theorems in Chapters V-VIII and the Appendices -- Referencing Conventions and Bibliography -- I. Referencing Conventions -- II. Abbreviations used in the Notes and the Bibliography -- III. Bibliography of Works Consulted: Ancient Authors -- IV. Modern Works: Books -- V. Modern Works: Articles -- Index of Names -- Index of Passages Cited from Ancient Works